📄 sfun.java
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package jmt.engine.math;
/*
* -------------------------------------------------------------------------
* $Id: Sfun.java,v 1.1.1.1 2005/06/06 07:43:35 Administrator Exp $
* -------------------------------------------------------------------------
* Copyright (c) 1997 - 1998 by Visual Numerics, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this software is freely
* granted by Visual Numerics, Inc., provided that the copyright notice
* above and the following warranty disclaimer are preserved in human
* readable form.
*
* Because this software is licenses free of charge, it is provided
* "AS IS", with NO WARRANTY. TO THE EXTENT PERMITTED BY LAW, VNI
* DISCLAIMS LEVEL_ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED
* TO ITS PERFORMANCE, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
* VNI WILL NOT BE LIABLE FOR ANY DAMAGES WHATSOEVER ARISING OUT OF THE USE
* OF OR INABILITY TO USE THIS SOFTWARE, INCLUDING BUT NOT LIMITED TO DIRECT,
* INDIRECT, SPECIAL, CONSEQUENTIAL, PUNITIVE, AND EXEMPLARY DAMAGES, EVEN
* IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.
*
* -------------------------------------------------------------------------
*/
/**
* Collection of special functions.
*/
public class Sfun {
/** The smallest relative spacing for doubles.*/
public final static double EPSILON_SMALL = 1.1102230246252e-16;
/** The largest relative spacing for doubles. */
public final static double EPSILON_LARGE = 2.2204460492503e-16;
// Series on [0,0.0625]
private static final double COT_COEF[] = {.240259160982956302509553617744970e+0, -.165330316015002278454746025255758e-1, -.429983919317240189356476228239895e-4, -.159283223327541046023490851122445e-6, -.619109313512934872588620579343187e-9, -.243019741507264604331702590579575e-11, -.956093675880008098427062083100000e-14, -.376353798194580580416291539706666e-16, -.148166574646746578852176794666666e-18};
// Series on the interval [0,1]
private static final double SINH_COEF[] = {0.1730421940471796, 0.08759422192276048, 0.00107947777456713, 0.00000637484926075, 0.00000002202366404, 0.00000000004987940, 0.00000000000007973, 0.00000000000000009};
// Series on [0,1]
private static final double TANH_COEF[] = {-.25828756643634710, -.11836106330053497, .009869442648006398, -.000835798662344582, .000070904321198943, -.000006016424318120, .000000510524190800, -.000000043320729077, .000000003675999055, -.000000000311928496, .000000000026468828, -.000000000002246023, .000000000000190587, -.000000000000016172, .000000000000001372, -.000000000000000116, .000000000000000009};
// Series on the interval [0,1]
private static final double ASINH_COEF[] = {-.12820039911738186343372127359268e+0, -.58811761189951767565211757138362e-1, .47274654322124815640725249756029e-2, -.49383631626536172101360174790273e-3, .58506207058557412287494835259321e-4, -.74669983289313681354755069217188e-5, .10011693583558199265966192015812e-5, -.13903543858708333608616472258886e-6, .19823169483172793547317360237148e-7, -.28847468417848843612747272800317e-8, .42672965467159937953457514995907e-9, -.63976084654366357868752632309681e-10, .96991686089064704147878293131179e-11, -.14844276972043770830246658365696e-11, .22903737939027447988040184378983e-12, -.35588395132732645159978942651310e-13, .55639694080056789953374539088554e-14, -.87462509599624678045666593520162e-15, .13815248844526692155868802298129e-15, -.21916688282900363984955142264149e-16, .34904658524827565638313923706880e-17};
// Series on the interval [0,0.25]
private static final double ATANH_COEF[] = {.9439510239319549230842892218633e-1, .4919843705578615947200034576668e-1, .2102593522455432763479327331752e-2, .1073554449776116584640731045276e-3, .5978267249293031478642787517872e-5, .3505062030889134845966834886200e-6, .2126374343765340350896219314431e-7, .1321694535715527192129801723055e-8, .8365875501178070364623604052959e-10, .5370503749311002163881434587772e-11, .3486659470157107922971245784290e-12, .2284549509603433015524024119722e-13, .1508407105944793044874229067558e-14, .1002418816804109126136995722837e-15, .6698674738165069539715526882986e-17, .4497954546494931083083327624533e-18};
// Series on the interval [0,1]
private static final double GAMMA_COEF[] = {.8571195590989331421920062399942e-2, .4415381324841006757191315771652e-2, .5685043681599363378632664588789e-1, -.4219835396418560501012500186624e-2, .1326808181212460220584006796352e-2, -.1893024529798880432523947023886e-3, .3606925327441245256578082217225e-4, -.6056761904460864218485548290365e-5, .1055829546302283344731823509093e-5, -.1811967365542384048291855891166e-6, .3117724964715322277790254593169e-7, -.5354219639019687140874081024347e-8, .9193275519859588946887786825940e-9, -.1577941280288339761767423273953e-9, .2707980622934954543266540433089e-10, -.4646818653825730144081661058933e-11, .7973350192007419656460767175359e-12, -.1368078209830916025799499172309e-12, .2347319486563800657233471771688e-13, -.4027432614949066932766570534699e-14, .6910051747372100912138336975257e-15, -.1185584500221992907052387126192e-15, .2034148542496373955201026051932e-16, -.3490054341717405849274012949108e-17, .5987993856485305567135051066026e-18, -.1027378057872228074490069778431e-18};
// Series for the interval [0,0.01]
private static final double R9LGMC_COEF[] = {.166638948045186324720572965082e0, -.138494817606756384073298605914e-4, .981082564692472942615717154749e-8, -.180912947557249419426330626672e-10, .622109804189260522712601554342e-13, -.339961500541772194430333059967e-15, .268318199848269874895753884667e-17};
// Series on [-0.375,0.375]
final private static double ALNRCS_COEF[] = {.103786935627437698006862677191e1, -.133643015049089180987660415531, .194082491355205633579261993748e-1, -.301075511275357776903765377766e-2, .486946147971548500904563665091e-3, -.810548818931753560668099430086e-4, .137788477995595247829382514961e-4, -.238022108943589702513699929149e-5, .41640416213865183476391859902e-6, -.73595828378075994984266837032e-7, .13117611876241674949152294345e-7, -.235467093177424251366960923302e-8, .425227732760349977756380529626e-9, -.771908941348407968261081074933e-10, .140757464813590699092153564722e-10, -.257690720580246806275370786276e-11, .473424066662944218491543950059e-12, -.872490126747426417453012632927e-13, .161246149027405514657398331191e-13, -.298756520156657730067107924168e-14, .554807012090828879830413216973e-15, -.103246191582715695951413339619e-15, .192502392030498511778785032449e-16, -.359550734652651500111897078443e-17, .672645425378768578921945742268e-18, -.126026241687352192520824256376e-18};
// Series on [0,1]
private static final double ERFC_COEF[] = {-.490461212346918080399845440334e-1, -.142261205103713642378247418996e0, .100355821875997955757546767129e-1, -.576876469976748476508270255092e-3, .274199312521960610344221607915e-4, -.110431755073445076041353812959e-5, .384887554203450369499613114982e-7, -.118085825338754669696317518016e-8, .323342158260509096464029309534e-10, -.799101594700454875816073747086e-12, .179907251139614556119672454866e-13, -.371863548781869263823168282095e-15, .710359900371425297116899083947e-17, -.126124551191552258324954248533e-18};
// Series on [0.25,1.00]
private static final double ERFC2_COEF[] = {-.69601346602309501127391508262e-1, -.411013393626208934898221208467e-1, .391449586668962688156114370524e-2, -.490639565054897916128093545077e-3, .715747900137703638076089414183e-4, -.115307163413123283380823284791e-4, .199467059020199763505231486771e-5, -.364266647159922287393611843071e-6, .694437261000501258993127721463e-7, -.137122090210436601953460514121e-7, .278838966100713713196386034809e-8, -.581416472433116155186479105032e-9, .123892049175275318118016881795e-9, -.269063914530674343239042493789e-10, .594261435084791098244470968384e-11, -.133238673575811957928775442057e-11, .30280468061771320171736972433e-12, -.696664881494103258879586758895e-13, .162085454105392296981289322763e-13, -.380993446525049199987691305773e-14, .904048781597883114936897101298e-15, -.2164006195089607347809812047e-15, .522210223399585498460798024417e-16, -.126972960236455533637241552778e-16, .310914550427619758383622741295e-17, -.766376292032038552400956671481e-18, .190081925136274520253692973329e-18};
// Series on [0,0.25]
private static final double ERFCC_COEF[] = {.715179310202924774503697709496e-1, -.265324343376067157558893386681e-1, .171115397792085588332699194606e-2, -.163751663458517884163746404749e-3, .198712935005520364995974806758e-4, -.284371241276655508750175183152e-5, .460616130896313036969379968464e-6, -.822775302587920842057766536366e-7, .159214187277090112989358340826e-7, -.329507136225284321486631665072e-8, .72234397604005554658126115389e-9, -.166485581339872959344695966886e-9, .401039258823766482077671768814e-10, -.100481621442573113272170176283e-10, .260827591330033380859341009439e-11, -.699111056040402486557697812476e-12, .192949233326170708624205749803e-12, -.547013118875433106490125085271e-13, .158966330976269744839084032762e-13, -.47268939801975548392036958429e-14, .14358733767849847867287399784e-14, -.444951056181735839417250062829e-15, .140481088476823343737305537466e-15, -.451381838776421089625963281623e-16, .147452154104513307787018713262e-16, -.489262140694577615436841552532e-17, .164761214141064673895301522827e-17, -.562681717632940809299928521323e-18, .194744338223207851429197867821e-18};
/**
* Private contructor, so nobody can make an instance of this class.
*/
private Sfun() {
}
/**
* Returns the inverse (arc) hyperbolic cosine of a double.
* @param x A double value.
* @return The arc hyperbolic cosine of x.
* If x is NaN or less than one, the result is NaN.
*/
static public double acosh(double x) {
double ans;
if (Double.isNaN(x) || x < 1) {
ans = Double.NaN;
} else if (x < 94906265.62) {
// 94906265.62 = 1.0/Math.sqrt(EPSILON_SMALL)
ans = Math.log(x + Math.sqrt(x * x - 1.0));
} else {
ans = 0.69314718055994530941723212145818 + Math.log(x);
}
return ans;
}
/**
* Returns the inverse (arc) hyperbolic sine of a double.
* @param x A double value.
* @return The arc hyperbolic sine of x.
* If x is NaN, the result is NaN.
*/
static public double asinh(double x) {
double ans;
double y = Math.abs(x);
if (Double.isNaN(x)) {
ans = Double.NaN;
} else if (y <= 1.05367e-08) {
// 1.05367e-08 = Math.sqrt(EPSILON_SMALL)
ans = x;
} else if (y <= 1.0) {
ans = x * (1.0 + csevl(2.0 * x * x - 1.0, ASINH_COEF));
} else if (y < 94906265.62) {
// 94906265.62 = 1/Math.sqrt(EPSILON_SMALL)
ans = Math.log(y + Math.sqrt(y * y + 1.0));
} else {
ans = 0.69314718055994530941723212145818 + Math.log(y);
}
if (x < 0.0) ans = -ans;
return ans;
}
/**
* Returns the inverse (arc) hyperbolic tangent of a double.
* @param x A double value.
* @return The arc hyperbolic tangent of x.
* If x is NaN or |x|>1, the result is NaN.
*/
static public double atanh(double x) {
double y = Math.abs(x);
double ans;
if (Double.isNaN(x)) {
ans = Double.NaN;
} else if (y < 1.82501e-08) {
// 1.82501e-08 = Math.sqrt(3.0*EPSILON_SMALL)
ans = x;
} else if (y <= 0.5) {
ans = x * (1.0 + csevl(8.0 * x * x - 1.0, ATANH_COEF));
} else if (y < 1.0) {
ans = 0.5 * Math.log((1.0 + x) / (1.0 - x));
} else if (y == 1.0) {
ans = x * Double.POSITIVE_INFINITY;
} else {
ans = Double.NaN;
}
return ans;
}
/**
* Returns the hyperbolic cosine of a double.
* @param x A double value.
* @return The hyperbolic cosine of x.
* If x is NaN, the result is NaN.
*/
static public double cosh(double x) {
double ans;
double y = Math.exp(Math.abs(x));
if (Double.isNaN(x)) {
ans = Double.NaN;
} else if (Double.isInfinite(x)) {
ans = x;
} else if (y < 94906265.62) {
// 94906265.62 = 1.0/Math.sqrt(EPSILON_SMALL)
ans = 0.5 * (y + 1.0 / y);
} else {
ans = 0.5 * y;
}
return ans;
}
/**
* Returns the cotangent of a double.
* @param x A double value.
* @return The cotangent of x.
* If x is NaN, the result is NaN.
*/
static public double cot(double x) {
double ans, ainty, ainty2, prodbg, y, yrem;
double pi2rec = 0.011619772367581343075535053490057; // 2/PI - 0.625
y = Math.abs(x);
if (y > 4.5036e+15) {
// 4.5036e+15 = 1.0/EPSILON_LARGE
return Double.NaN;
}
// Carefully compute
// Y * (2/PI) = (AINT(Y) + REM(Y)) * (.625 + PI2REC)
// = AINT(.625*Y) + REM(.625*Y) + Y*PI2REC = AINT(.625*Y) + Z
// = AINT(.625*Y) + AINT(Z) + REM(Z)
ainty = (int) y;
yrem = y - ainty;
prodbg = 0.625 * ainty;
ainty = (int) prodbg;
y = (prodbg - ainty) + 0.625 * yrem + y * pi2rec;
ainty2 = (int) y;
ainty = ainty + ainty2;
y = y - ainty2;
int ifn = (int) (ainty % 2.0);
if (ifn == 1) y = 1.0 - y;
if (y == 0.0) {
ans = Double.POSITIVE_INFINITY;
} else if (y <= 1.82501e-08) {
// 1.82501e-08 = Math.sqrt(3.0*EPSILON_SMALL)
ans = 1.0 / y;
} else if (y <= 0.25) {
ans = (0.5 + csevl(32.0 * y * y - 1.0, COT_COEF)) / y;
} else if (y <= 0.5) {
ans = (0.5 + csevl(8.0 * y * y - 1.0, COT_COEF)) / (0.5 * y);
ans = (ans * ans - 1.0) * 0.5 / ans;
} else {
ans = (0.5 + csevl(2.0 * y * y - 1.0, COT_COEF)) / (0.25 * y);
ans = (ans * ans - 1.0) * 0.5 / ans;
ans = (ans * ans - 1.0) * 0.5 / ans;
}
if (x != 0.0) ans = sign(ans, x);
if (ifn == 1) ans = -ans;
return ans;
}
/*
* Evaluate a Chebyschev series
*/
static double csevl(double x, double coef[]) {
double b0, b1, b2, twox;
int i;
b1 = 0.0;
b0 = 0.0;
b2 = 0.0;
twox = 2.0 * x;
for (i = coef.length - 1; i >= 0; i--) {
b2 = b1;
b1 = b0;
b0 = twox * b1 - b2 + coef[i];
}
return 0.5 * (b0 - b2);
}
/*
* Correction term used by logBeta.
*/
private static double dlnrel(double x) {
double ans;
if (x <= -1.0) {
ans = Double.NaN;
} else if (Math.abs(x) <= 0.375) {
ans = x * (1.0 - x * Sfun.csevl(x / .375, ALNRCS_COEF));
} else {
ans = Math.log(1.0 + x);
}
return ans;
}
/**
* Returns the error function of a double.
* @param x A double value.
* @return The error function of x.
*/
static public double erf(double x) {
double ans;
double y = Math.abs(x);
if (y <= 1.49012e-08) {
// 1.49012e-08 = Math.sqrt(2*EPSILON_SMALL)
ans = 2 * x / 1.77245385090551602729816748334;
} else if (y <= 1) {
ans = x * (1 + csevl(2 * x * x - 1, ERFC_COEF));
} else if (y < 6.013687357) {
// 6.013687357 = Math.sqrt(-Math.getLog(1.77245385090551602729816748334 * EPSILON_SMALL))
ans = sign(1 - erfc(y), x);
} else {
ans = sign(1, x);
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